Section G3: Differential Amplifiers

Transcription

1 Scton G3: Dffrntal Amplfrs h dffrntal amplfr may b mplmntd usng Js or Fs and s a commonly usd buldng block n analog dsgn. W ar gong to b concntratng on th J mplmntaton of th dffrntal par as mttr-coupld, common-mttr (or mttr-rsstor) amplfrs. n addton to prodng th nput stag of th opratonal amplfr, ths confguraton s th foundaton for mttr-coupld logc (L), a ry hgh spd, unrsally mployd famly of logc crcuts. h smplst form of th dffrntal amplfr s shown to th rght (a modfd rson of Fgur 9. n your txt). hs crcut s formd usng two matchd transstors (Q and Q ), n th confguraton whos mttrs ha bn td togthr. h dffrntal par has two nputs ( and ), and thr possbl outputs ( o, o, and out ). A coupl of nots about ths crcut: h output oltag, out, s th dffrnc btwn o and o, or out o o. h notaton of th mttr rsstor has bn changd to snc s common to both amplfrs. As w wll s shortly, ths rsstor may b th quat rsstanc of a currnt sourc as dscussd n th prous scton, both to sa chp spac and rduc fabrcaton complxty (transstors ar smallr than rsstors and t s asr to mak a whol bunch of th sam thng). Howr t s cratd, t s ncssary that ha a larg alu to kp th oltag drop across t narly constant wth rasonabl changs n currnt. h rsstors n th crcut abo may b rplacd wth othr transstors n som applcatons (onc agan, to rmo th rqurmnt of rsst componnts). t s absolutly mprat howr, that whatr s n th collctor crcut nsurs that Q and Q nr ntr saturaton. o sol for th dc transfr charactrstcs of th dffrntal amplfr, w bgn by usng th xprsson for bas-mttr oltag drd from

2 quaton 4.0. n trms of total nstantanous oltag and collctor currnt, and, w now ha, whr o s th rrs saturaton currnt and w ha onc agan assumd that n. Usng ths rlatonshp, and nglctng bas currnts so that, w can xprss and as ;. (quaton 9., orrctd) Now, wrtng th KL quaton about th bas juncton loop as ndcatd by th rd path n th fgur abo, w ha 0 +. (quaton 9., Modfd) Assumng w ha matchd dcs, and ar th sam for Q and Q. Substtutng th nformaton of quaton 9. nto quaton 9., rarrangng and mployng th proprty of logarthms [(A)-()(A/)], + / / 0;. (quaton 9.3, Modfd) akng th xponntal of both sds of th last quaton abo, w can sol for th currnt rato, / : and or ) ( ;. (quaton 9.4, Modfd) A KL at th coupld mttrs of Q and Q ylds

3 +. (quaton 9.5) Substtutng th nformaton from th modfd rson of quaton 9.4 nto quaton 9.5, w can fnd fnal xprssons for and n trms of crcut and physcal paramtrs. Not that n th quatons blow, and. call howr that th mttr and collctor currnts ar actually rlatd by α, whr αβ/(β+) for β>>. + xp + xp ( [ ( ) / ] [ ) / ]. (quaton 9.6) Lt s look at what th rsults of quaton 9.6 tll us. For a gn currnt, th amplfr rsponds only to th dffrnc btwn th oltags appld to th bass of Q and Q (.., and n Fgur 9. and n th quatons abo). hs s th foundaton of dffrntal amplfr opraton (and s whr th nam coms from). f c (w ll talk about th c and d notaton n a coupl of mnuts), / and th currnt s splt nly btwn th two transstors for any alu of common-mod oltag,. f th dffrntal oltag, d -, bcoms gratr than about 4 ( 00m), th currnt wll flow almost ntrly n on of th two transstors. Spcfcally, o f d >00m, approxmatly 98% of wll flow through Q and Q wll b ssntally cut off, or o f d <-00m, approxmatly 98% of wll flow through Q and Q wll b ssntally cut off. N: h 4 fgur coms from th ntrscton of plots of th mttr currnts of th two transstors or by sog th xprssons of quaton 9.6. Lnar amplfcaton usng th dffrntal par s only possbl for small dffrntal oltags. Gnrally, nputs of on th ordr of (approxmatly 0m) ar usd as a gud to nsur ar bhaors, but your author stats that a dffrnc of lss than 00m s an adquat rstrcton. Not that, for th smpl common-mttr confguratons, 00m may b too larg (othr sourcs ct a lmt n th rang of /). Howr, to ncras th rang of ar opraton, small mttr rsstors may b addd ths s probably whr your author gts hs lmt.

4 am gong to tak a slghtly dffrnt approach that your author n th nxt sgmnt of ths dscusson, but wll attmpt to kp all notatons th sam. Not that n all of th followng analyss w wll assum that r o s ry larg so that t may b nglctd unlss spcfcally statd othrws. Dffrntal Mod Gan For dffrntal-mod opraton, w apply an ac dffrntal oltag, d, btwn th bass of Q and Q, n addton to a dc common mod oltag. From th symmtry of th dffrntal par, ths dffrntal nput sgnal should dd nly btwn th bas mttr junctons of th two transstors. n trms of th dffrntal and common-mod oltags, th orgnal nput oltags of Fgur 9. may b xprssd as d d + and. (quaton 9.9, Modfd) Not that - d, whch s what w want. hs s tru snc th nput to Q s qual n magntud to that of Q, but 80 o out of phas (ndcatd by th ngat sgn). h small sgnal modl of th dffrntal amplfr n dffrntal mod s llustratd n Fgur 9.a and s rproducd to th rght. Usng th symmtry of th dffrntal par and comparng th notaton of ths crcut wth Fgur 9., w ha out od and o o od /. Not that although o and o ar of qual magntud, thy ar 80 o out of phas (.., o - o ). Modfyng quaton 9.6 to rflct th dffrntal nput oltag d, + xp + xp [ / ] [ / ] d d. Analyzng th abo rlatonshps, w can s that as th currnt on transstor ncrass, th currnt through th othr transstor dcrass at th sam rat. hrfor, snc th total currnt through nr changs,

5 th oltag chang across s zro. hs mans that th ac sgnal oltag across s zro and th rsstor may b rplacd wth a short crcut n th ac quat, as shown n th abo fgur. f th transstors ar basd at an mttr currnt of /, and wth ffctly shortd for ac opraton, anothr way of lookng at th currnt ncras/dcras may b dlopd n trms of r, whr r βr, assumng β>>. A KL n thr bas loop ylds d r βr b b r. Sog for, w gt an xprsson for th currnt ncras/dcras to b d. Assumng, th output at ach collctor trmnal wll b 80 o r out of phas wth a magntud of od d gm d o o, r or, th oltag masurd btwn th two collctor trmnals wll b g. d od r m d As ndcatd abo, th output of th dffrntal amplfr may b takn thr dffrntally (or doubl-ndd), whr th output s takn btwn th two collctors, or sngl-ndd, whr th output s takn from thr collctor to ground. f th output s takn btwn th collctor trmnals, th dffrntal gan (dffrntal, or doubl-ndd output) of th amplfr wll b qual to A d gm. (quaton 9.) r As statd arlr, th sngl-ndd output oltags at o and o ar of qual magntud, and ar ach on-half th magntud of th doubl-ndd output oltag, od. f th output s takn btwn thr collctor trmnal and ground, th dffrntal gan (sngl-ndd output) of th amplfr wll b qual to on-half of th gan calculatd for th doubl-ndd output, or

6 A d. (quaton 9.) r f th dffrntal amplfr has rsstors n th nddual mttr lgs, th doubl-ndd dffrntal gan s A d, r + whch should b xpctd, bcaus w now ha coupld mttr-rsstor () amplfrs. h nput rsstanc n dffrntal mod s th quat rsstanc sn btwn th two bass. f thr s no rsstor n th mttr lgs of th crcuts, and s ffctly shortd, w can s from Fgur 9.a that th nput rsstanc wll b n ( dffrntal mod). (quaton 9.6) r For th coupld confguraton (.., wth s n th mttr lads), w can mploy th mpdanc rflcton rul to xprss th nput rsstanc as n( dffrntal mod) r + ( β + ) ( β + )( r + ). ommon Mod Gan h md-frquncy ac small sgnal modl for common mod opraton s gn n Fgur 9.b and s rproducd blow and to th lft. A smplfd rson of ths small sgnal modl s gn blow and to th rght. Not that th common mod crcut may b splt nto a par of paralll crcuts, calld th common-mod half-crcut, wth / and wth an mttr rsstor of (rcall that, so w han t changd th orgnal crcut of Fgur 9.).

7 h nput to ach half crcut s of th sam magntud and phas. f th transstors ar matchd, th crcut s prfctly symmtrc, and w assum that >>r, w may dr th sngl-ndd common-mod gan, usng thr half crcut, to b A. (quaton 9.3) dally, th dffrntal (doubl-ndd) common-mod gan s zro snc - c - c 0 (and th crcut s prfctly symmtrc). Practcally, howr, crcuts ar not prfctly symmtrc and dcs cannot b prfctly matchd. hs mans that th common-mod gan wll not b zro for a dffrntal output. h doubl-ndd output oltag may thn b xprssd n trms of th common-mod and dffrntal-mod gans as follows: + out Ad d + Ac c Ad ( ) + Ac. (quaton 9.4) n f som dscrpancy from dal xsts n th dffrntal amplfr crcut, th doubl-ndd common-mod gan s much smallr than that for th sngl-ndd output. hrfor, th nput stag of a multstag amplfr (an op-amp for xampl) s usually a balancd dffrntal par wth th output takn dffrntally (doubl-ndd). hs mans that th dffrntal amplfr stag wll ha a low common-mod gan, whch w wll s blow s a dsrabl qualty n trms of th common-mod rjcton rato (th nxt part talks about th M, hold on).

8 h common-mod nput rsstanc s found usng th amplfr of Fgur 9.b (th complt small sgnal modl to th lft abo). h quat rsstanc sn by th common-mod sourcs s calculatd by rflctng ( ) up nto th bas crcut, and s gn by n ( common mod) r β,(quaton 9.7, Modfd) + ( β + ) whr th fnal smplfcaton s mad by assumng β>> and >>r. Not that th trm β may bcom ry larg. f ths occurs, th transstor output rsstanc may no longr b nglctd n a paralll rlatonshp (.., r o s not much largr than β ) and th common mod nput rsstanc bcoms ro n( common mod) β. As usual, f th assumpton β>> dos not hold, all β trms bcom β+. Not that th nput rsstanc for common-mod opraton s much largr than that for dffrntal-mod opraton. ommon-mod jcton ato (M) An ffct dffrntal par wll ract prmarly to th dffrnc btwn th nput oltags and th dffrntal-mod gan wll b much largr than th common-mod gan. A fgur of mrt for th dffrntal amplfr s calld th common-mod rjcton rato, or M. h M s dfnd as th rato of th dffrntal-mod gan to th common-mod gan. Usng th xprssons dfnd for sngl-ndd outputs, th M s gn by A d M A c / r / r g m. Normally, th M s xprssd n d as M 0 log 0 log( gm ) r. (quaton 9.5) Dffrntal Amplfr wth onstant urrnt Sourc For an dal dffrntal amplfr, th dffrntal common-mod gan would b zro and th M would b nfnt. Snc ths s not practcal, on way

9 to ncras th M s to mak as larg as possbl. h fabrcaton of larg rsstancs on an s not practcal, so an altrnat approach nols rplacng wth a transstor confgurd as a dc currnt sourc. h closr th currnt sourc bhas to an dal currnt sourc, th hghr th common-mod rjcton rato (rcall that an dal currnt sourc has nfnt mpdanc). h crcut of Fgur 9.3a (rproducd to th rght) s an xampl of a dffrntal amplfr wth a dod-compnsatd fxd-bas currnt sourc. h dod compnsaton maks th crcut opraton lss dpndnt on tmpratur aratons (s Scton D0 for a dscusson of dod compnsaton n J crcuts). h dod D and transstor Q 3 ar slctd so that thy ha narly dntcal charactrstcs or th rang of opratng tmpraturs. Just for fun ralz that D may b anothr J that s matchd to Q 3 and s dod-connctd (ts collctor s td to ts bas). Also, any of th currnt sourcs w dscussd n th prous scton may b usd to crat th constant currnt sourc usd for basng purposs. o analyz th crcut of Fgur 9.3a and dtrmn th M, w nd to calculat th quat rsstanc of th constant currnt sourc crcut (calld H by your author). A modfd rson of Fgur 9.3b s prsntd to th rght and wll b usd for ths analyss. Not that ha changd th dsgnatons of th nods and oltags across r o and snc th and n Fgur 9.3b of your txt ar not th sam as ndcatd n Fgur 9.3a. Also, th H of Fgur 9.3b s th sam currnt as n Fgur 9.3a (.., th collctor currnt of Q 3 ). Wth all ths n mnd, th crcut to th rght s th small sgnal quat of th currnt sourc (wth th bas, collctor and mttr of Q 3 notd by 3, 3 and 3, rspctly) f th dod s assumd to b dal and β>>. Snc t s xpctd that th quat rsstanc of ths currnt sourc wll b larg, w must also nclud th output rsstanc, r o, n our analyss. Fnally, by assumng an dal dod, s our old frnd.

10 h quat rsstanc of th constant currnt sourc crcut s gn by H A. (quaton 9.8, Modfd) + H Wrtng a KL quaton at nod 3, w ha + H A β b, (quaton 9.0, Modfd) ro whl a KL quaton at nod 3 ylds A β (quaton 9., Modfd) r o xprssng th oltags n trms of th currnts and H, w ha A ( H β ) ro. (quaton 9., Modfd) ( r + ) m not gong to go through all th algbra hr, but f you substtut quaton 9. nto 9., and sol for H, thn substtut ths rsult (along wth quaton 9.) nto quaton 9.8, you gt an xprsson for H to b H r + + r o [ + ( r + ) / ] + ( r + ) / + β r o. (quaton 9.4) o smplfy ths nasty lookng xprsson, your author maks a srs of assumptons: o mantan bas stablty, w us 0.β. β>>, so that >>/β r o >> 0. >> r Puttng all ths nto quaton 9.4 ylds th ncrdbly smpl rsult H r o, (quaton 9.3) whr r o A /. Not that all of th abo approxmatons must b ald for quaton 9.3 to b usd. f t can b, lf s good, H s ndpndnt of β and s hug!

11 Dffrntal Amplfr wth Sngl-ndd nput and utput Fgur 9.4 (gn to th rght) llustrats a dffrntal amplfr whr th nput to Q s groundd and th output s takn as o. As dscussd n th prous scton, a constant currnt sourc s usd n plac of. hs confguraton s known as a sngdd nput and output amplfr wth phas rrsal. o analyz ths amplfr, all w ha to do s st 0 n th arlr quatons. f w assum that th quat rsstanc of th currnt sourc s ry larg, th common-mod gan s approxmatly qual to zro. hs mans th sngl-ndd dffrntal-mod gan of th amplfr wll dtrmn th output, whch wll b out Ad d. (quaton 9.33) r h ngat sgn ndcats that thr s a 80 o phas shft btwn th nput ( ) and th output ( o ), as xpctd for a common mttr amplfr. f a phas rrsal s not dsrd for a sngl-ndd output that s rfrncd to ground, th output can b takn from Q.